
Using a coherent superposition of boundary spin networks, defined as an infinite series over the spins controlled by couplings dual to the spins, actually amounts to considering a generating function for the spin network evaluations. This is the same logic as for a simple quantum harmonic oscillator, in which matrix elements of an operator in the coherent state basis. That is
\(\langle z|\hat{{\mathcal {O}}}| \tilde{z}\rangle =\sum _{n,m\in {\mathbb {N}}}{\sqrt{n!m!}} \langle n|\hat{{\mathcal {O}}}| m\rangle \) 
can be understood as a generating function for the matrix elements \(\langle n|\hat{{\mathcal {O}}}| m\rangle \)  controlled by the complex couplings \(z\)  and \(\tilde{z}\) .
Here, we focus on 3-valent boundary graphs, for which we don't need intertwiner labels, so that spin network evaluations \({\mathcal {E}}_{\Gamma }[\lbrace j_{l},\iota _{n}\rbrace _{l,n\in \Gamma }]\)  simply depend on the spins on the boundary graph links. We will simply write \({\mathcal {E}}_{\Gamma }[\lbrace j_{l}\rbrace _{l\in \Gamma }]\) . Then we define coherent spin network evaluations similarly as for the harmonic oscillator as:
\(E_{\Gamma }[\lbrace Y_{l}\rbrace _{l\in \Gamma }]=\sum _{\lbrace j_{l}\in 2\rbrace } Y_{l}^{2j_{l}} \,{\mathcal {W}}[\lbrace j_{l}\rbrace _{l\in \Gamma }]\,{\mathcal {E}}_{\Gamma }[\lbrace j_{l}\rbrace _{l\in \Gamma }]\,,\) 
with the couplings \(Y_{l}\in {\mathbb {C}}\)  and weights \({\mathcal {W}}[\lbrace j_{l}\rbrace _{l\in \Gamma }]\)  possibly involving factorials of the spins [1]}, [2]}, [3]}.
Generating functions is a powerful mathematical tool. For instance, they typically map the asymptotic behaviour, here at large spins i.e. the semi-classical regime for length scales very large compared to the Planck length, onto poles of the generating function.

The coherent spin superpositions, or equivalently the generating functions, that we consider here allow for exact analytical resummation of the Ponzano-Regge amplitudes as rational functions in the couplings. They are actually the generalization of Schwinger's generating function for the \(\lbrace 6j\rbrace \) -symbols [4]}, [5]}, they were introduced as coherent spin network states in [6]}, [7]}, [8]} and showed to lead to exact closed formula for spinfoam models in [9]}, [10]}. At the end, the evaluations \(E_{\Gamma }[\lbrace Y_{l}\rbrace _{l\in \Gamma }]\) , for specific well-chosen weights \({\mathcal {W}}[\lbrace j_{l}\rbrace _{l\in \Gamma }]\) , were shown to be given by the inverse squared partition function of the 2D Ising model with inhomogeneous couplings \(\tan ^{-1} Y_{l}\)  on the boundary graph [3]}.
For instance, the generating function for the \(\lbrace 6j\rbrace \) -symbols corresponds to the inverse of the square of the 2D Ising model on the tetrahedron with six variables \(Y_{1,..,6}\)  living on the edges and representing the strength of the coupling between the four triangles [12]}.
Then, in general for arbitrary 3-cells and their boundary graphs, this provides formulas for the Ponzano-Regge amplitude as holomorphic functions of couplings living on the boundary of the 3-cells.
These formulas are at the heart of the proposed holographic duality between 3D quantum gravity defined by the Ponzano-Regge path integral and the 2D Ising model [3]}.

